The first phase of the temperature control lab is to derive a dynamic model of the system with guess values for parameters. The three important elements for a control loop are the measurement device (thermistor temperature sensor), an actuator (transistor), and capability to perform computerized control (USB interface). At maximum output the transistor dissipates 1 W of power at 100% heater output. The mass of the transistor and heat sink with fins is 4 gm.

Steel has a heat capacity of 500 `J/{kg K}`. The surface area of the heater and sensor is about 12 `cm^2`. A convective heat transfer coefficient for quiescent air is approximately 10 `W/{m^2K}`. The heat generated by the transistor transfers away from the device primarily by convection but radiative heat transfer may also be a contributing factor. The radiative heat transfer can be included in the model to determine what fraction of heat is lost by convection and heat radiation. Heat transfer is improved with a thermal coupling (white epoxy) that connects the two components.

Quantity

Value

Initial temperature (T0)

296.15 K (23oC)

Ambient temperature (`T_\infty`)

296.15 K (23oC)

Heater output (Q)

0 to 1 W

Heater factor (`\alpha`)

0.01 W/(% heater)

Heat capacity (Cp)

500 J/kg-K

Surface Area (A)

1.2x10-3 m2 (12 cm2)

Mass (m)

0.004 kg (4 gm)

Overall Heat Transfer Coefficient (U)

10 W/m2-K

Emissivity (`\epsilon`)

0.9

Stefan Boltzmann Constant (`\sigma`)

5.67x10-8 W/m2-K4

Create a dynamic model of the dynamic response between input power to the transistor and the temperature sensed by the thermistor. Use an energy balance to start the derivation.

$$m\,c_p\frac{dT}{dt} = \sum \dot h_{in} - \sum \dot h_{out} + Q$$

Expand or simplify terms that are needed for this application. The full energy balance includes convection and radiation terms.

where `m` is the mass, `c_p` is the heat capacity, `T` is the temperature, `U` is the heat transfer coefficient, `A` is the area, `T_\infty` is the ambient temperature, `\epsilon=0.9` is the emissivity, `\sigma =` 5.67x10-8 `W/{m^2 K^4}` is the Stefan-Boltzmann constant, and `Q` is the percentage heater output. The parameter `\alpha` is a factor that relates heater output (0-100%) to power dissipated by the transistor in Watts. Use this equation to develop a dynamic simulation of the temperature response due to an impulse (off, on, off) in the heater output. Leave the heater on for sufficient time to observe nearly steady state conditions.

Investigate issues such as whether radiative heat transfer is significant, is the temperature response inherently first order or higher order, and what values of uncertain parameters in the physics based model help the predicted temperature agree with the data.